Numerical modelling of unsteady mixed flows

This thesis concerns mixed flows (which are characterized by the simultaneous occurrence of free-surface and pressurized flow in sewers, tunnels, culverts or under bridges), and contributes to the improvement of the existing numerical tools for modelling these phenomena. The classic Preissmann slot approach is selected due to its simplicity and capability of predicting results comparable to those of a more recent and complex two-equation model, as shown here with reference to a laboratory test case. In order to enhance the computational efficiency, a local time stepping strategy is implemented in a shock-capturing Godunov-type finite volume numerical scheme for the integration of the de Saint-Venant equations. The results of different numerical tests show that local time stepping reduces run time significantly (between −29% and −85% CPU time for the test cases considered) compared to the conventional global time stepping, especially when only a small region of the flow field is surcharged, while solution accuracy and mass conservation are not impaired. The second part of this thesis is devoted to the modelling of the hydraulic effects of potentially pressurized structures, such as bridges and culverts, inserted in open channel domains. To this aim, a two-dimensional mixed flow model is developed first. The classic conservative formulation of the 2D shallow water equations for free-surface flow is adapted by assuming that two fictitious vertical slots, normally intersecting, are added on the ceiling of each integration element. Numerical results show that this schematization is suitable for the prediction of 2D flooding phenomena in which the pressurization of crossing structures can be expected. Given that the Preissmann model does not allow for the possibility of bridge overtopping, a one-dimensional model is also presented in this thesis to handle this particular condition. The flows below and above the deck are considered as parallel, and linked to the upstream and downstream reaches of the channel by introducing suitable internal boundary conditions. The comparison with experimental data and with the results of HEC-RAS simulations shows that the proposed model can be a useful and effective tool for predicting overtopping and backwater effects induced by the presence of bridges and culverts.

Canonical solution groups of a homomorphism : manipulator kinematics, nonparametric regression and distributed object systems

The kinematic mapping of a rigid open-link manipulator is a homomorphism between Lie groups. The homomorphisrn has solution groups that act on an inverse kinematic solution element. A canonical representation of solution group operators that act on a solution element of three and seven degree-of-freedom (do!) dextrous manipulators is determined by geometric analysis. Seven canonical solution groups are determined for the seven do! Robotics Research K-1207 and Hollerbach arms. The solution element of a dextrous manipulator is a collection of trivial fibre bundles with solution fibres homotopic to the Torus. If fibre solutions are parameterised by a scalar, a direct inverse funct.ion that maps the scalar and Cartesian base space coordinates to solution element fibre coordinates may be defined. A direct inverse pararneterisation of a solution element may be approximated by a local linear map generated by an inverse augmented Jacobian correction of a linear interpolation. The action of canonical solution group operators on a local linear approximation of the solution element of inverse kinematics of dextrous manipulators generates cyclical solutions. The solution representation is proposed as a model of inverse kinematic transformations in primate nervous systems. Simultaneous calibration of a composition of stereo-camera and manipulator kinematic models is under-determined by equi-output parameter groups in the composition of stereo-camera and Denavit Hartenberg (DH) rnodels. An error measure for simultaneous calibration of a composition of models is derived and parameter subsets with no equi-output groups are determined by numerical experiments to simultaneously calibrate the composition of homogeneous or pan-tilt stereo-camera with DH models. For acceleration of exact Newton second-order re-calibration of DH parameters after a sequential calibration of stereo-camera and DH parameters, an optimal numerical evaluation of DH matrix first order and second order error derivatives with respect to a re-calibration error function is derived, implemented and tested. A distributed object environment for point and click image-based tele-command of manipulators and stereo-cameras is specified and implemented that supports rapid prototyping of numerical experiments in distributed system control. The environment is validated by a hierarchical k-fold cross validated calibration to Cartesian space of a radial basis function regression correction of an affine stereo model. Basic design and performance requirements are defined for scalable virtual micro-kernels that broker inter-Java-virtual-machine remote method invocations between components of secure manageable fault-tolerant open distributed agile Total Quality Managed ISO 9000+ conformant Just in Time manufacturing systems.

Mathematical and numerical modelling of dispersion-managed solitons, autosolitons and self-similar optical pulses

This thesis presents theoretical investigation of three topics concerned with nonlinear optical pulse propagation in optical fibres. The techniques used are mathematical analysis and numerical modelling. Firstly, dispersion-managed (DM) solitons in fibre lines employing a weak dispersion map are analysed by means of a perturbation approach. In the case of small dispersion map strengths the average pulse dynamics is described by a perturbation approach (NLS) equation. Applying a perturbation theory, based on the Inverse Scattering Transform method, an analytic expression for the envelope of the DM soliton is derived. This expression correctly predicts the power enhancement arising from the dispersion management.Secondly, autosoliton transmission in DM fibre systems with periodical in-line deployment of nonlinear optical loop mirrors (NOLMs) is investigated. The use of in-line NOLMs is addressed as a general technique for all-optical passive 2R regeneration of return-to-zero data in high speed transmission system with strong dispersion management. By system optimisation, the feasibility of ultra-long single-channel and wavelength-division multiplexed data transmission at bit-rates ³ 40 Gbit s-1 in standard fibre-based systems is demonstrated. The tolerance limits of the results are defined.Thirdly, solutions of the NLS equation with gain and normal dispersion, that describes optical pulse propagation in an amplifying medium, are examined. A self-similar parabolic solution in the energy-containing core of the pulse is matched through Painlevé functions to the linear low-amplitude tails. The analysis provides a full description of the features of high-power pulses generated in an amplifying medium.

Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices

A numerical continuation method has been carried out seeking solutions for two distinct flow configurations, planar Couette flow (PCF) and laterally heated flow in a vertical slot (LHF). We found that the spanwise vortex solution in LHF identifies a new solution in PCF. The vortical structure of our new solution has the shape of a hairpin observed ubiquitously in high-Reynolds-number turbulent flow, and we believe this discovery may provide the paradigm for a hierarchical organization of coherent structures in turbulent shear layers.

Total solution of Gibilaro and Rowe model for a segregating fluidised bed

This study re-examines the one-dimensional equilibrium model of Gibilaro and Rowe (1974) for a segregating gas fluidized bed. The model was based on volumetric jetsam concentration and divided the bed contents into bulk and wake phases, taking account of bulk and wake flux, segregation, exchange between the bulk and wake phases, and axial mixing. Due to the complex nature of the model and its unstable solution, the lack of computing power at the time prevented the authors from doing little more than the analytical solutions to specific cases of this model. This paper provides a numerical total solution and allows the effect of the respective parameters to be compared for the first time. There is also a comparison with experimental results, which showed a reasonable agreement.

Numerical simulation of corneal transport processes

This paper presents a numerical study on the transport of ions and ionic solution in human corneas and the corresponding influences on corneal hydration. The transport equations for each ionic species and ionic solution within the corneal stroma are derived based on the transport processes developed for electrolytic solutions, whereas the transport across epithelial and endothelial membranes is modelled by using phenomenological equations derived from the thermodynamics of irreversible processes. Numerical examples are provided for both human and rabbit corneas, from which some important features are highlighted.

On the Operational Solution of a System of Fractional Differential Equations

MSC 2010: 26A33, 44A45, 44A40, 65J10

Numerical and experimental analysis of sensitivity-enhanced RI sensor based on Ex-TFG in thin cladding fiber

We report a highly sensitive refractive index (RI) sensor in the aqueous solution, which is based on an 81°-tilted fiber grating structure inscribed into a thin cladding fiber with 40 μm cladding radius. The numerical analysis has indicated that the RI sensitivity of cladding resonance mode of the grating can be significantly enhanced with reducing cladding size. This has been proved by the experimental results as the RI sensitivities of TM and TE resonance peaks in the index region of 1.345 have been increased to 1180 nm/RIU and 1150 nm/RIU, respectively, from only 200 and 170 nm/RIU for the same grating structure inscribed in standard telecom fiber with 62.5-μm cladding radius. Although the temperature sensitivity has also increased, the change in temperature sensitivity is still insignificant in comparison with RI sensitivity enhancement.

Explicit Solution of Bitsadze-Samarskii Problem

Иван Димовски, Юлиан Цанков - В статията е намерено точно решение на задачата на Бицадзе-Самрски (1) за уравнението на Лаплас, като е използвано операционно смятане основано на некласическа двумернa конволюция. На това точно решение може да се гледа като начин за сумиране на нехармоничния ред по синуси на решението, получен по метода на Фурие.

Exact solution for the energy spectrum of Kelvin-wave turbulence in superfluids

We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at near-zero temperatures. In this paper, we show analytically that the solution proposed by [L’vov and Nazarenko, JETP Lett. 91, 428 (2010)] enjoys existence, uniqueness, and regularity of the prefactor. Furthermore, we present numerical results of the dynamical equation that describes to leading order the nonlocal regime of the Kelvin-wave dynamics. We compare our findings with the analytical results from the proposed local and nonlocal theories for Kelvin-wave dynamics and show an agreement with the nonlocal predictions. Accordingly, the spectrum proposed by L’vov and Nazarenko should be used in future theories of quantum turbulence. Finally, for weaker wave forcing we observe an intermittent behavior of the wave spectrum with a fluctuating dissipative scale, which we interpreted as a finite-size effect characteristic of mesoscopic wave turbulence.

Solution of the Least Squares Method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima.

Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure

The focus of this work is to develop and employ numerical methods that provide characterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies.

We first propose the fabric tensor formalism to describe the structure and evolution of lithium-ion electrode microstructure during the calendaring process. Fabric tensors are directional measures of particulate assemblies based on inter-particle connectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, we show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode.

We then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. We explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, we discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh discretizations, we find that employing randomness can improve convergence behavior and provide a computational savings.

The Thick Level-Set model is implemented to describe brittle media undergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severity of degradation and uses a length scale to limit the damage gradient. In terms of energy dissipated by fracture and mean fragment size, we find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies.

Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the battery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. We find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems.

The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing.

Solitons in Exciton-Polariton System: Reduced Modelling, Analysis, and Numerical Studies

In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.

First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system,

and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons.

Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons.

Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively.

Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses

and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital

stability but no $L_2$ stability.

Temperature fields in a channel partially filled with a porous material under local thermal non-equilibrium condition-an exact solution

This work examines analytically the forced convection in a channel partially filled with a porous material and subjected to constant wall heat flux. The Darcy–Brinkman–Forchheimer model is used to represent the fluid transport through the porous material. The local thermal non-equilibrium, two-equation model is further employed as the solid and fluid heat transport equations. Two fundamental models (models A and B) represent the thermal boundary conditions at the interface between the porous medium and the clear region. The governing equations of the problem are manipulated, and for each interface model, exact solutions, for the solid and fluid temperature fields, are developed. These solutions incorporate the porous material thickness, Biot number, fluid to solid thermal conductivity ratio and Darcy number as parameters. The results can be readily used to validate numerical simulations. They are, further, applicable to the analysis of enhanced heat transfer, using porous materials, in heat exchangers.

Enhancing the numerical performance of fire field models (CMS Paper)

This paper describes two new techniques designed to enhance the performance of fire field modelling software. The two techniques are "group solvers" and automated dynamic control of the solution process, both of which are currently under development within the SMARTFIRE Computational Fluid Dynamics environment. The "group solver" is a derivation of common solver techniques used to obtain numerical solutions to the algebraic equations associated with fire field modelling. The purpose of "group solvers" is to reduce the computational overheads associated with traditional numerical solvers typically used in fire field modelling applications. In an example, discussed in this paper, the group solver is shown to provide a 37% saving in computational time compared with a traditional solver. The second technique is the automated dynamic control of the solution process, which is achieved through the use of artificial intelligence techniques. This is designed to improve the convergence capabilities of the software while further decreasing the computational overheads. The technique automatically controls solver relaxation using an integrated production rule engine with a blackboard to monitor and implement the required control changes during solution processing. Initial results for a two-dimensional fire simulation are presented that demonstrate the potential for considerable savings in simulation run-times when compared with control sets from various sources. Furthermore, the results demonstrate the potential for enhanced solution reliability due to obtaining acceptable convergence within each time step, unlike some of the comparison simulations.